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A Reissner-Nordstrom solution with non-vanishing fermion number
Volume 130B, number 3,4
PHYSICS LETTERS
20 October 1983
A R E I S S N E R - N O R D S T R O M SOLUTION WITH NON-VANISHING FERMION NUMBER
Daksh LOHIYA Tata Institute of Fundamental Research, Homt Bhabha Road, Bombay 400 005, Indta Received 12 April 1983 Revlsed manuscript recewed 30 June 1983
This commumcatlon reports on a search for a zero frequency solution of the Dtrac operator m the presence of black hole dyons. The presence of such modes leads to a solunon with a non-vamshmg fermlon number. Such solutions are expected to be Important m dlscussslons of chtral symmetry breaking processes near black holes. These features are intimately related to recent assertions of monopole catalysed proton decay processes.
1. Introduction. The possibility of "monopole catalysed proton decay" [1 ] has triggered a lot of interest in the study of the effect of light fermions on the charge degrees of freedom of magnetic monopoles. According to the anomalous divergence of the axial vector current [2], ( ~buj5u) = 2irn0qJ75o/+ (aO/4~r)euvopFaUVFff p ,
(1.1)
even for massless fermions, chirality would be changing with time. (Here ,I~ is an isospinor and Fff v are the generalized gauge fields - a being the lsospin index). Blaer et al. [3 ] have shown that in circumstances as existing in the presence of dyons, e~apFUVF~P is non-vanishing and the system radiates fermions. Callan [1 ] has shown that the dyon instability can be expressed in terms of the Interaction of the massless fermlons with the charge degree of freedom of magnetic monopoles. The dyon electric field is completely screened and the monopole gets surrounded by "halo" of a chlral symmetry breaking condensate of fermions. Such configurations with a non-vanishing F * F interpolate between different chiral states of a monopole. From the properties of the Dirac operator m ~he field of these Interpolating configurations, ~t follows that definite fermion isospIn states can exist only as ingoing or outgoing states of different chirahty. It is this aspect that gives rise to proton decay when we consider a fundamental monopole sitting in an SU(2) 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
subgroup of an SU(5) grand unified theory. Under this SU(2), the u I and u 2 quarks ( 1 , 2 , 3 being colour indices) form isospin down parts of two left-handed Fermi fields, the d 3 quark and the positron, on the other hand, form the isospin up parts of two righthanded Fermi fields. We therefore have processes with u 1 + u 2 in the mmal state and d3 + e+ in the final state. This would lead to a proton (Ul, u 2, d 3 singlet) decaying into a pion (d3d3) and a positron. An equivalent study of the dyon instability is possible by looking for zero energy modes for the Dirac operator in the field of the dyon. These are the so-called Jackiw-Rebbl modes [4]. The existence of such nondegenerate, isolated, zero-energy solutions to the Dirac equation in the presence of a soliton (monopole) amplies that a sohton has a fermionic structure, namely, 1 that of a degenerate doublet with fermlon number + 7If one now considers a scattering problem of fennions from the sohton [3], the energy spectrum of emitted fermions shows a resonant enhancement at zero total energy. That is, it is the Jacklw-Rebbl mode which acquires a width where the dyon becomes unstable. The production of chiral charge, consistent with an axial anomaly equation is curved space-time, has recently been demonstrated [5] for dyon solutions to (a) Einstein-Maxwell and (b) Emsteln-Yang-Mdls equations. Both the solutions are described by the metric
179
Volume 130B, number 3,4
PHYSICS LETTERS
ds 2 = (A/r 2) dt 2 _ (A/r2)- 1 dr 2 - r2(sin20 dq02 + d 0 2 ) ,
(1.2)
with A -~ r 2 -- 2 M r + p 2 + Q2 = (r - r+)(r - r ) .
In the first e x a m p l e , P and Q are, respectively, the magnetic and electric charges of a Reissner-Nordstrom solution. There exists a gauge in which the vector potential may be locally written as A = (Q/r) d t - P cos 0 d ¢ .
(1.3)
20 October 1983
search for zero frequency solutions of the Dirac operator in the above background. This study is the subject of this article. 2. Consider the Dirac equation for a massless, charged spinor field in the two component formalism, The Dlrac spinor q~ is expressed as a column vector [~A,]. p A and QA' are two component "undashed" and "dashed" spinors (the right-handed and the lefthanded parts, respectively, of the four-spinor q~. The Dirac equation takes the form: (~TAA, -- l e A A A , ) P A = ( V AA, + l e A A A , ) Q A = 0 .
(2.1)
The second example is a spherically symmetric solutton of a coupled SO(3) Yang-Mflls-Higgs system in curved space-tn-ne. The system is described by the Lagrange density
Expressing this in terms of the spin coefficients and the directional derivatives of the standard N e w m a n Penrose formalism gives
Z? = X / ~
(D + e - ~ - i eV)P 0
[ - R / 1 6 7 r G - 4-1-1:ap° -l:a -pa
_ ~gUV ~ u c a (-/)yea _ XV(~)] ,
(1.4)
with
+ (6" + I1 - a - i e f 2 ' ) P ' = 0 , (6 + t3 - r - ief2)P 0
=
a _
+
~uA v
-
b
geabcAuAv
c
, + ( A + I~ - - 7 - - i e V ' ) P '
(2.2a)
= 0
q) udpa =- 3#q9 a + g e a b c A b ~ c ,
(D + e - p - ieV)~) 0' .v(¢)
=
_
0
z/a)
Ca is a Higgs isotnplet, its vacuum expectation value being/a2p,. The solution is described by the metric (1.2), the gauge fields being given by At = g - l ( ~ + q / r ) ~ a ,
A7 = _g-leai/fl
~a = +012/,A)l/2~a ,
+ (8 + t3 - r - i e ~ ) 0 l ' = 0 , (6* +/3* - r* - i e ~ ' ) 0 0' + (A +/a*
,
(1.5)
_
7*
_
xeV')
~91'
= 0,
(2.2b)
where the spmor components of the vector potential are defmed by
with Q = q / g , P = 1/g and P = arbitrary constant. Performing a gauge transformation
(A,I)=V,
(A,n}=V';
A p ~ UAa#oaU - 1 + ( 1 / g ) U ~ # U - 1 ,
(A,m)= ~,
(A, N) = ~2',
with
the null tetrads [l, n, m, r~] being chosen as:
U = exp{i(o3~/2 + o 2 0 / 2 ) ) ,
l = (r2/A) 3 / 3 t + 3 / 3 r ,
reduces the field configuration to a smgle component
m = (1/X/2 r ) [ ~ / 3 0 + (i/sin 0) 3/3~] .
A t * = g - l ( ~ + q / r ) o 3 d t - g - 1 o3d~0
of an abelian dyon with magnetic strength 1]g and electric charge q/g. The instability of the fermiomc vacuum suggests a 180
n --~ - '
(2.3)
- ½(alr2)
l
r
(2.4)
The non-vanishmg spin coefficients turn out to be = -1/r,
/3 = -c~ = - c o t ( 0 ) / 2 x Q r
P = - A / 2 r 3 , P - 7 = 014 - r ) / 2 r 2 .
(2.5)
To look for static (zero frequency) solutions, the angular and radial dependence may be separated by:
pO = ( f l / r ) exp(lrn¢) , 0 1 ' =gl exp(im~o) , 0 0, = (g2/r) exp(im~o),
p t =f2 e x p ( l m ¢ ) ,
(2.6)
[
,flTeam~=_[eP+l/2YlmR-1/2 ~ f2-J t_eP-112Ylm R+l/2_] ' gl q l m ~ _ ~ e P + l / 2 Y l m R u g ~ g2-J e * =LeP- 1/2 Ylm R-1/23 "
I
(2.7)
R ~ (s = -+ ½) satisfy
A-s+l(d/dr)[ks+ldR/dr]
20 October 1983
PHYSICS LETTERS
Volume 130B. number 3.4
+ {k 2 + 2iks(r - M)
A2Z = V Z .
(2.11)
Now to look for modes which possess the symmetry of the background field viz. the Reissner-Nordstrom solution, we consider the spherically symmetric (,4 = 0) modes. For these, eq. (2.11) is simple to solve [6] gwing Y ~ exp 0 e Qr*). Thus the zero frequency mode is bounded but not normahsable. Bounded and normalizable modes are however obtainable if the Dlrac field has a mass. For example, if we have massless isospinor Dirac field coupling to a Hlggs field, then the vacuum expectation value [eq. (1.5)] of the Higgs field induces an effective mass for both lsospin components of the fermmn. The Dirac equation for the zero frequency spherically symmetric case, reduces to:
(ieQ/A1/2)X' + r-l(d/dr)[A1/2X'] + 2iseaA + [A + (s +~)] A}R = 0 ,
(2.8)
(ieQ/A1/2)X - r-l(d/dr)[A1/2X]
where
k -- - e Q r ,
(ieQ/A1/2)~ " - r-l(d/dr)[A1/2y]
d / d r ~ = (A/r) d/dr
[,,o,,,1,01,, 00,]
-
=
[r_/(r+
[r+/(r+
-
-
Y+QY=0,
with
(2.12)
r
y
-
-¢
=- [eP+l/2YlnX,eP-1/2Yln X ;eP-1/2 lnY, eP+l/2Yln Y ], (2.9)
Further we introduce Y1/2 = A1/2R1/2, Y-l~2 = R 1/7, and the operators A+ - d/dr* + ieQ, A 2 = A A+. This reduces (2.8) to A2y+5~A
-+ K Y = 0
where
.
r_)] In(r - r+)
r _ ) ] ln(r - r _ ) .
+-K X = O,
( i e Q / & l / 2 ) Y ' + r - l ( d / d r ) [ A 1 / 2 y '] T-K Y ' = O,
A =- (l + ½)2 _ e2p2 .
T]cds can be reduced to a real effective potential equation (9). To this effect, one introduces a new radial coordinate r* def'med by:
r*
-T-K X ' = 0),
and K = g times the expectation value of the Higgs field. The _+signs before K in eq. (2.12) depend on the isospin up or down cases respectwely. The solutions can be written immediately. With Q = 0, the solutions go as
(2.10)
( ; rdrl 1/2) exp + K d A I / 2 ] . r
X'~(1/rA
P÷
= (d/d r*) [ln(r/A 1/2)] , Q = A Air 2 .
with
For solutions to be normalizable, they should not blow up faster than A -1/2 at the honzon, The above solution has this behavmur and also goes to zero at When the charge on the dyon is non-vanishing the solution bounded at the horizon goes as
q± = + 2ieQA1/2 ,
(1 -- r+/r) -1/2 exp(-- X/~ r* + leQr*)
and
at mfimty. Thus the solutions shall have normalizable zero modes. The existence of these bound states imply that one should not take the viewpoint that the Reissner-
Defining Y =: (A1/2/r)q+Z + 2ieQA+Z,
17+_= A A/r 2 ¥ A ll2(dldr*)(A1/2/r), reduces (2.10) to
181
Volume 130B, number 3,4
PHYSICS LETTERS
Nordstrom solution exists independent of the fermions which then bind to it with zero energy. Rather we should say that the R e i s s n e r - N o r d s t r o m solution is doubly degenerate and processes a fermlon number. Because o f the existence o f such zero modes, the femlion vacuum around the black hole m o n o p o l e ts ill defined. If the fermion is an isodoublet under the SU(2) subgroup o f SU(5) which mixes the quark and the lepton sectors then the existence o f the zero mode implies a resonant absorption and reemisslon probability for an S-wave. Now the physical distinction between the quark and the lepton sectors is done with respect to the local SU(2) frame of the Higgs field. This frame is ambiguous near the horizon of the black hole. Thus fermion doublets would lose their identity if they manage to reach the horizon. Hence the existence o f zero modes would be responsible for p r o t o n decays if for example the SU(2) multiplets (d~) and (u, fi) are considered together [7]. The existence of fermion zero modes has led to a conjecture on the emergence of a dynamical supersymmetry in monopoles. Indeed, black hole solutions with a fermion number were previously found in members of supermultiplets of black hole sohtonic states m 0 ( 2 ) and 0 ( 3 ) supergravlty theories [8]. The precise correspondence is not yet clear. The instability o f the fermionic vacuum in the pres-
182
20 October 1983
ence of a magnetic field has also been observed by Vilenkin [9]. Our analysis would therefore imply the presence o f zero frequency eigenmodes of the Dirac operator in Vilenkin's case. The author would like to thank Professor S.W. Hawking for suggesting the problem and for discussions.
References [1] V.A. Rubakov, Nucl. Phys. B203 (1982) 311, C.G. Callan, Dyon-Fermlon dynamics, Princeton Umversity preprint (1982). [2] J.S. Bell and R. Jacklw, Nuovo Clmento 60A (1969) 47, S.L. Adler, Phys. Rev. 177 (1969) 2426. [3] A.S. Blaer, N.H. Christ and J.F. Tang, Phys. Rev. Lett. 47 (1981) 1364; Phys. Rev. D25 (1982) 2128. [4] R. Jacklw and C. Rebbl, Phys. Rev. D13 (1976) 3396. [5] D. Lohiya, Ann. Phys. 145 (1983); DAMTP Ph.D. Thesis. Some aspects of particle creation from charged black holes (1982). [6] S. Chandrasekhar, m' General relativity, eds. S.W. Hawking and W. Israel (Cambridge U.P., London, 1979), S. Chandrasekhar and S. Detweiler, Proc. R. Soc. (London) A352 (1976) 325. [7] Y. Nambu, Magnetic monopoles and related topics, talk Topical Symp. (Tokyo, 1982). [8] G.W. Gibbons, Sohton states and central charges in extended supergravlty theories, DAMTP preprmt (1982). [9] A. Vilenkin, Phys. Rev. D22 (1981) 3080.
PHYSICS LETTERS
20 October 1983
A R E I S S N E R - N O R D S T R O M SOLUTION WITH NON-VANISHING FERMION NUMBER
Daksh LOHIYA Tata Institute of Fundamental Research, Homt Bhabha Road, Bombay 400 005, Indta Received 12 April 1983 Revlsed manuscript recewed 30 June 1983
This commumcatlon reports on a search for a zero frequency solution of the Dtrac operator m the presence of black hole dyons. The presence of such modes leads to a solunon with a non-vamshmg fermlon number. Such solutions are expected to be Important m dlscussslons of chtral symmetry breaking processes near black holes. These features are intimately related to recent assertions of monopole catalysed proton decay processes.
1. Introduction. The possibility of "monopole catalysed proton decay" [1 ] has triggered a lot of interest in the study of the effect of light fermions on the charge degrees of freedom of magnetic monopoles. According to the anomalous divergence of the axial vector current [2], ( ~buj5u) = 2irn0qJ75o/+ (aO/4~r)euvopFaUVFff p ,
(1.1)
even for massless fermions, chirality would be changing with time. (Here ,I~ is an isospinor and Fff v are the generalized gauge fields - a being the lsospin index). Blaer et al. [3 ] have shown that in circumstances as existing in the presence of dyons, e~apFUVF~P is non-vanishing and the system radiates fermions. Callan [1 ] has shown that the dyon instability can be expressed in terms of the Interaction of the massless fermlons with the charge degree of freedom of magnetic monopoles. The dyon electric field is completely screened and the monopole gets surrounded by "halo" of a chlral symmetry breaking condensate of fermions. Such configurations with a non-vanishing F * F interpolate between different chiral states of a monopole. From the properties of the Dirac operator m ~he field of these Interpolating configurations, ~t follows that definite fermion isospIn states can exist only as ingoing or outgoing states of different chirahty. It is this aspect that gives rise to proton decay when we consider a fundamental monopole sitting in an SU(2) 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
subgroup of an SU(5) grand unified theory. Under this SU(2), the u I and u 2 quarks ( 1 , 2 , 3 being colour indices) form isospin down parts of two left-handed Fermi fields, the d 3 quark and the positron, on the other hand, form the isospin up parts of two righthanded Fermi fields. We therefore have processes with u 1 + u 2 in the mmal state and d3 + e+ in the final state. This would lead to a proton (Ul, u 2, d 3 singlet) decaying into a pion (d3d3) and a positron. An equivalent study of the dyon instability is possible by looking for zero energy modes for the Dirac operator in the field of the dyon. These are the so-called Jackiw-Rebbl modes [4]. The existence of such nondegenerate, isolated, zero-energy solutions to the Dirac equation in the presence of a soliton (monopole) amplies that a sohton has a fermionic structure, namely, 1 that of a degenerate doublet with fermlon number + 7If one now considers a scattering problem of fennions from the sohton [3], the energy spectrum of emitted fermions shows a resonant enhancement at zero total energy. That is, it is the Jacklw-Rebbl mode which acquires a width where the dyon becomes unstable. The production of chiral charge, consistent with an axial anomaly equation is curved space-time, has recently been demonstrated [5] for dyon solutions to (a) Einstein-Maxwell and (b) Emsteln-Yang-Mdls equations. Both the solutions are described by the metric
179
Volume 130B, number 3,4
PHYSICS LETTERS
ds 2 = (A/r 2) dt 2 _ (A/r2)- 1 dr 2 - r2(sin20 dq02 + d 0 2 ) ,
(1.2)
with A -~ r 2 -- 2 M r + p 2 + Q2 = (r - r+)(r - r ) .
In the first e x a m p l e , P and Q are, respectively, the magnetic and electric charges of a Reissner-Nordstrom solution. There exists a gauge in which the vector potential may be locally written as A = (Q/r) d t - P cos 0 d ¢ .
(1.3)
20 October 1983
search for zero frequency solutions of the Dirac operator in the above background. This study is the subject of this article. 2. Consider the Dirac equation for a massless, charged spinor field in the two component formalism, The Dlrac spinor q~ is expressed as a column vector [~A,]. p A and QA' are two component "undashed" and "dashed" spinors (the right-handed and the lefthanded parts, respectively, of the four-spinor q~. The Dirac equation takes the form: (~TAA, -- l e A A A , ) P A = ( V AA, + l e A A A , ) Q A = 0 .
(2.1)
The second example is a spherically symmetric solutton of a coupled SO(3) Yang-Mflls-Higgs system in curved space-tn-ne. The system is described by the Lagrange density
Expressing this in terms of the spin coefficients and the directional derivatives of the standard N e w m a n Penrose formalism gives
Z? = X / ~
(D + e - ~ - i eV)P 0
[ - R / 1 6 7 r G - 4-1-1:ap° -l:a -pa
_ ~gUV ~ u c a (-/)yea _ XV(~)] ,
(1.4)
with
+ (6" + I1 - a - i e f 2 ' ) P ' = 0 , (6 + t3 - r - ief2)P 0
=
a _
+
~uA v
-
b
geabcAuAv
c
, + ( A + I~ - - 7 - - i e V ' ) P '
(2.2a)
= 0
q) udpa =- 3#q9 a + g e a b c A b ~ c ,
(D + e - p - ieV)~) 0' .v(¢)
=
_
0
z/a)
Ca is a Higgs isotnplet, its vacuum expectation value being/a2p,. The solution is described by the metric (1.2), the gauge fields being given by At = g - l ( ~ + q / r ) ~ a ,
A7 = _g-leai/fl
~a = +012/,A)l/2~a ,
+ (8 + t3 - r - i e ~ ) 0 l ' = 0 , (6* +/3* - r* - i e ~ ' ) 0 0' + (A +/a*
,
(1.5)
_
7*
_
xeV')
~91'
= 0,
(2.2b)
where the spmor components of the vector potential are defmed by
with Q = q / g , P = 1/g and P = arbitrary constant. Performing a gauge transformation
(A,I)=V,
(A,n}=V';
A p ~ UAa#oaU - 1 + ( 1 / g ) U ~ # U - 1 ,
(A,m)= ~,
(A, N) = ~2',
with
the null tetrads [l, n, m, r~] being chosen as:
U = exp{i(o3~/2 + o 2 0 / 2 ) ) ,
l = (r2/A) 3 / 3 t + 3 / 3 r ,
reduces the field configuration to a smgle component
m = (1/X/2 r ) [ ~ / 3 0 + (i/sin 0) 3/3~] .
A t * = g - l ( ~ + q / r ) o 3 d t - g - 1 o3d~0
of an abelian dyon with magnetic strength 1]g and electric charge q/g. The instability of the fermiomc vacuum suggests a 180
n --~ - '
(2.3)
- ½(alr2)
l
r
(2.4)
The non-vanishmg spin coefficients turn out to be = -1/r,
/3 = -c~ = - c o t ( 0 ) / 2 x Q r
P = - A / 2 r 3 , P - 7 = 014 - r ) / 2 r 2 .
(2.5)
To look for static (zero frequency) solutions, the angular and radial dependence may be separated by:
pO = ( f l / r ) exp(lrn¢) , 0 1 ' =gl exp(im~o) , 0 0, = (g2/r) exp(im~o),
p t =f2 e x p ( l m ¢ ) ,
(2.6)
[
,flTeam~=_[eP+l/2YlmR-1/2 ~ f2-J t_eP-112Ylm R+l/2_] ' gl q l m ~ _ ~ e P + l / 2 Y l m R u g ~ g2-J e * =LeP- 1/2 Ylm R-1/23 "
I
(2.7)
R ~ (s = -+ ½) satisfy
A-s+l(d/dr)[ks+ldR/dr]
20 October 1983
PHYSICS LETTERS
Volume 130B. number 3.4
+ {k 2 + 2iks(r - M)
A2Z = V Z .
(2.11)
Now to look for modes which possess the symmetry of the background field viz. the Reissner-Nordstrom solution, we consider the spherically symmetric (,4 = 0) modes. For these, eq. (2.11) is simple to solve [6] gwing Y ~ exp 0 e Qr*). Thus the zero frequency mode is bounded but not normahsable. Bounded and normalizable modes are however obtainable if the Dlrac field has a mass. For example, if we have massless isospinor Dirac field coupling to a Hlggs field, then the vacuum expectation value [eq. (1.5)] of the Higgs field induces an effective mass for both lsospin components of the fermmn. The Dirac equation for the zero frequency spherically symmetric case, reduces to:
(ieQ/A1/2)X' + r-l(d/dr)[A1/2X'] + 2iseaA + [A + (s +~)] A}R = 0 ,
(2.8)
(ieQ/A1/2)X - r-l(d/dr)[A1/2X]
where
k -- - e Q r ,
(ieQ/A1/2)~ " - r-l(d/dr)[A1/2y]
d / d r ~ = (A/r) d/dr
[,,o,,,1,01,, 00,]
-
=
[r_/(r+
[r+/(r+
-
-
Y+QY=0,
with
(2.12)
r
y
-
-¢
=- [eP+l/2YlnX,eP-1/2Yln X ;eP-1/2 lnY, eP+l/2Yln Y ], (2.9)
Further we introduce Y1/2 = A1/2R1/2, Y-l~2 = R 1/7, and the operators A+ - d/dr* + ieQ, A 2 = A A+. This reduces (2.8) to A2y+5~A
-+ K Y = 0
where
.
r_)] In(r - r+)
r _ ) ] ln(r - r _ ) .
+-K X = O,
( i e Q / & l / 2 ) Y ' + r - l ( d / d r ) [ A 1 / 2 y '] T-K Y ' = O,
A =- (l + ½)2 _ e2p2 .
T]cds can be reduced to a real effective potential equation (9). To this effect, one introduces a new radial coordinate r* def'med by:
r*
-T-K X ' = 0),
and K = g times the expectation value of the Higgs field. The _+signs before K in eq. (2.12) depend on the isospin up or down cases respectwely. The solutions can be written immediately. With Q = 0, the solutions go as
(2.10)
( ; rdrl 1/2) exp + K d A I / 2 ] . r
X'~(1/rA
P÷
= (d/d r*) [ln(r/A 1/2)] , Q = A Air 2 .
with
For solutions to be normalizable, they should not blow up faster than A -1/2 at the honzon, The above solution has this behavmur and also goes to zero at When the charge on the dyon is non-vanishing the solution bounded at the horizon goes as
q± = + 2ieQA1/2 ,
(1 -- r+/r) -1/2 exp(-- X/~ r* + leQr*)
and
at mfimty. Thus the solutions shall have normalizable zero modes. The existence of these bound states imply that one should not take the viewpoint that the Reissner-
Defining Y =: (A1/2/r)q+Z + 2ieQA+Z,
17+_= A A/r 2 ¥ A ll2(dldr*)(A1/2/r), reduces (2.10) to
181
Volume 130B, number 3,4
PHYSICS LETTERS
Nordstrom solution exists independent of the fermions which then bind to it with zero energy. Rather we should say that the R e i s s n e r - N o r d s t r o m solution is doubly degenerate and processes a fermlon number. Because o f the existence o f such zero modes, the femlion vacuum around the black hole m o n o p o l e ts ill defined. If the fermion is an isodoublet under the SU(2) subgroup o f SU(5) which mixes the quark and the lepton sectors then the existence o f the zero mode implies a resonant absorption and reemisslon probability for an S-wave. Now the physical distinction between the quark and the lepton sectors is done with respect to the local SU(2) frame of the Higgs field. This frame is ambiguous near the horizon of the black hole. Thus fermion doublets would lose their identity if they manage to reach the horizon. Hence the existence o f zero modes would be responsible for p r o t o n decays if for example the SU(2) multiplets (d~) and (u, fi) are considered together [7]. The existence of fermion zero modes has led to a conjecture on the emergence of a dynamical supersymmetry in monopoles. Indeed, black hole solutions with a fermion number were previously found in members of supermultiplets of black hole sohtonic states m 0 ( 2 ) and 0 ( 3 ) supergravlty theories [8]. The precise correspondence is not yet clear. The instability o f the fermionic vacuum in the pres-
182
20 October 1983
ence of a magnetic field has also been observed by Vilenkin [9]. Our analysis would therefore imply the presence o f zero frequency eigenmodes of the Dirac operator in Vilenkin's case. The author would like to thank Professor S.W. Hawking for suggesting the problem and for discussions.
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